Method 1: Global import

Download full repo (dist + src) from:
https://gitee.com/layui/layui

Place it into:
public/libs/layui

Add to public/assets/js/require-backend.js:

require.config({
  paths: {
    layui: "../libs/layui/src/layui",
  },
  shim: {
    layui: {
      deps: ["css!../libs/layui/src/css/layui.css"],
      init: function () {
        console.log("init layui");
        return this.layui.config({ dir: "/assets/libs/layui/src/" });
      },
    },
  },
});

Target page usage:

define(["jquery", "bootstrap", "backend", "form", "table", "layui"], function ($, undefined, Backend, Form, Table, layui) {
  var Controller = {
    index: function () {
      layui.use(["form", "upload"], function (form, upload) {});
    },
  };
  return Controller;
});

Pack/minify before use (then clear browser cache):

php think min -m all -r all

Method 2: Dynamic import

In public/assets/js/backend-init.js:

define(['backend'], function (Backend) {
  require.config({
    paths: { 'layui': '../libs/layui/src/layui' },
    shim: { 'layui': { deps: ['css!../libs/layui/src/css/layui.css'], init: function () { return this.layui.config({ dir: '/assets/libs/layui/src/' }); } } }
  })
});

Target page usage with dynamic load:

define(["jquery", "bootstrap", "backend", "form", "table", "layui"], function ($, undefined, Backend, Form, Table, layui) {
  var Controller = {
    index: function () {
      require(["layui"], function () {
        layui.use(["form", "upload"], function (form, upload) {});
      });
    },
  };
  return Controller;
});

Pack again when done:

php think min -m all -r all